# Alternative Constructions of the Reals

I have been reading about the construction of the reals (Tao's Analysis I, Classic Set Theory by Golderi, and bits of extra info from Naive Set Theory by Halmos and Rudin's Principles of Mathematical Analysis) starting from the axiom of infinity, and working up to define $$\mathbb{N}$$, $$\mathbb{Z}$$, $$\mathbb{Q}$$ and then $$\mathbb{R}$$. I understand this construction and I like it in that it gives us the nice hierarchy of naturals to integers to rationals and eventually to the reals.

However, I've noticed that many details of the construction were complicated by the introduction of 0 in $$\mathbb{N}$$, and the introduction of negative numbers when constructing $$\mathbb{Z}$$. For example, we need to be mindful of division by zero when defining $$\mathbb{Q}$$, and negative numbers make multiplication annoying (like in the definition of multiplication if we construct reals as Dedekind left sets).

Therefore, I was wondering whether this alternative process might be a bit better:

1. Start with the Peano axioms, and going with the convention of taking 1 as our starting point instead of 0. The number 0 (the additive identity) will only be introduced near the end of our construction.
2. Define $$\mathbb{Q}^+$$ using the relation $$(a, b) \sim (c, d) \iff ad = bc$$. This relation should be an equivalence relation, and we can then use it to define each positive rational as an equivalence class of ordered pairs. Order can be defined easily using $$[(a, b)] \leq [(c, d)] \iff bc \leq ad$$. Addition and multiplication can also be defined to follow our intuitions of rationals (e.g. $$[(a, b)] \times [(c, d)] = [(ac, bd)])$$.
3. Now construct $$\mathbb{R}^+$$. Defining each positive real as a Dedekind left set should work, and this will give us the least upper bound existence property.
4. We now finally introduce 0 and the negative numbers using the relation $$(a, b) \sim (c, d) \iff a + d = b + c$$. The members of $$\mathbb{R}$$ are now equivalence classes of ordered pairs of positive reals. The process is similar to defining the integers right after the naturals in the 'standard' construction. It gets a bit annoying because 0 is now $$[(x, x)]$$ instead of $$[(0, 0)]$$ so some proofs might be slightly more cumbersome, but overall this allows us to postpone 0 and negatives right till the very end so I think it's a worthwhile tradeoff.

I was wondering whether anyone has any comments about this strategy? Have I missed anything? I've been trying it out during my spare time and it seems to be holding up so far. Does anyone know any books that use this construction?

At the end of the day, I am aware that it doesn't quite matter how we perform the construction, so this is really just a matter of curiosity for me. I'm also aware that my strategy doesn't immediately give us the $$\mathbb{N}$$, $$\mathbb{Z}$$, $$\mathbb{Q}$$, $$\mathbb{R}$$ hierarchy so it's not perfect either. But it does seem to align better with the historical development of numbers, since (if I'm not mistaken) mathematicians accepted fractions and decimals before negative numbers. In any case, please let me know what you think and thanks in advance for your input!

• Yea this works. I've done this out before out of curiosity. It's a bit painful to establish order properties with multiplication, and the distributive property. As per your last paragraph, negative numbers were developed before $\sqrt2$ was found by Pythagoras, so... Commented Jan 8, 2022 at 4:50
• @RushabhMehta Your remark concerning negative numbers is very interesting! Can you give a reference where the occured for the first time? Commented Jan 8, 2022 at 23:40
• @KritikerderElche After a bit of reading, the picture is a bit more complicated. Pythagoras and his fellow geometers didn't view $\sqrt2$ as a number, but rather as a measurement for which there was no number. It was Eudoxus of Cnidus who is credited as the one who found a way of dealing with these sorts of measurements, but he too didn't hold the view of these lengths corresponding to numbers. And thus it was Euclid who first treated irrational numbers in the Western tradition, as far as I am aware. This came after the first usages of negative numbers in 200 BC. Commented Jan 8, 2022 at 23:50
• @KritikerderElche However, this isn't the full story. Knowledge of irrational numbers as well as the treatment of them as numbers was argued to be known to Indian mathematicians as early as 800 BC, far earlier than even the geometers under Pythagoras. However, this is still somewhat disputed, so the claim is true or false, depending on your position on this issue. Commented Jan 8, 2022 at 23:50
• My point of view is that we never worry about constructions while working with real numbers: it is their properties rather than definitions that matter. But your interesting construction do have some pedagogical values, provided that for some reasons a lecturer is willing to introduce definitions in a “non-standard” order. Commented Jan 9, 2022 at 7:16